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(m-n)6-8m3

2 min read Jun 16, 2024
(m-n)6-8m3

(m-n)^6-8m^3: Factoring and Simplifying

This expression involves factoring a difference of cubes and then simplifying the resulting expression. Let's break down the steps:

Step 1: Recognizing the Difference of Cubes Pattern

The expression (m-n)^6 - 8m^3 fits the pattern of a difference of cubes:

  • a^3 - b^3 = (a-b)(a^2 + ab + b^2)

In this case:

  • a = (m-n)^2
  • b = 2m

Step 2: Applying the Difference of Cubes Formula

Applying the formula, we get:

(m-n)^6 - 8m^3 = [(m-n)^2 - 2m][(m-n)^4 + 2m(m-n)^2 + 4m^2]

Step 3: Simplifying Further

  • Simplify the first factor: (m-n)^2 - 2m = m^2 - 2mn + n^2 - 2m
  • Expand the second factor: (m-n)^4 + 2m(m-n)^2 + 4m^2 = m^4 - 4m^3n + 6m^2n^2 - 4mn^3 + n^4 + 2m^3 - 4m^2n + 2mn^2 + 4m^2

Final Factored Expression

The fully factored and simplified expression is:

(m^2 - 2mn + n^2 - 2m)[m^4 - 2m^3n + 6m^2n^2 - 4mn^3 + n^4 + 2m^3 - 4m^2n + 2mn^2 + 4m^2]

This expression can't be factored further without using complex numbers. Therefore, this is the simplest form of the given expression.

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